Method of generating space-time codes for generalized layered space-time architectures

ABSTRACT

Space-time codes for use with layered architectures with arbitrary numbers of antennas are provided such as rate k/n convolutional codes (e.g., rates higher than or equal to 1/n where n is the number of transmit antennas). Convolutional codes for layered space-time architectures are generated using matrices over the ring F[[D]] of formal power series in variable D.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of provisional U.S. applicationSerial No. 60/153,936, filed Sep. 15, 1999.

Related subject matter is disclosed in U.S. patent application Ser. No.09/397,896, filed Sep. 17, 1999, and U.S. patent application Ser. No.09/613,938 of Hesham El Gamal et al for “System Employing ThreadedSpace-Time Architecture for Transporting Symbols and Receivers forMulti-User Detection and Decoding of Symbols”, filed even date herewith,the entire contents of both said applications being expresslyincorporated herein by reference.

FIELD OF THE INVENTION

The invention relates generally to generating codes for use in layeredspace-time architectures.

BACKGROUND OF THE INVENTION

Unlike the Gaussian channel, the wireless channel suffers frommulti-path fading. In such fading environments, reliable communicationis made possible only through the use of diversity techniques in whichthe receiver is afforded multiple replicas of the transmitted signalunder varying channel conditions.

Recently, information theoretic studies have shown that spatialdiversity provided by multiple transmit and/or receive antennas allowsfor a significant increase in the capacity of wireless communicationsystems operated in Rayleigh fading environment. Two approaches forexploiting this spatial diversity have been proposed. In the firstapproach, channel coding is performed across the spatial dimension, aswell as the time, to benefit from the spatial diversity provided byusing multiple transmit antennas. The term “space-time codes” is used torefer to this coding scheme. One potential drawback of this schemes isthat the complexity of the maximum likelihood (ML) decoder isexponential in the number of transmit antennas. A second approach reliesupon a layering architecture at the transmitter and signal processing atthe receiver to achieve performance asymptotically close to the outagecapacity. In this “layered” space-time architecture, no attempt is madeto optimize the channel coding scheme. Further, conventional channelcodes are used to minimize complexity. Accordingly, a need exists for alayering architecture, signal processing, and channel coding that aredesigned and optimized jointly.

SUMMARY OF THE INVENTION

The disadvantages of existing channel coding methods and receivers formultiple antenna communication systems are overcome and a number ofadvantages are realized by the present invention which providesspace-time codes for use in layered architectures having arbitrarynumbers of antennas and arbitrary constellations. Algebraic designs ofspace-time codes for layered architectures are provided in accordancewith the present invention.

In accordance with an aspect of the present invention, rate k/nconvolutional codes are provided for layered space-time architectures(e.g., rates higher than or equal to 1/n where n is the number oftransmit antennas).

In accordance with another aspect of the present invention,convolutional codes for layered space-time architectures are generatedusing matrices over the ring F[[D]] of formal power series in variableD.

BRIEF DESCRIPTION OF THE DRAWINGS

The various aspects, advantages and novel features of the presentinvention will be more readily comprehended from the following detaileddescription when read in conjunction with the appended drawing, inwhich:

FIG. 1 is a block diagram of a multiple antenna wireless communicationsystem constructed in accordance with an embodiment of the presentinvention.

Throughout the drawing figures, like reference numerals will beunderstood to refer to like parts and components.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 1. Space-TimeSignaling

FIG. 1 depicts a multiple antenna communication system 10 with ntransmit antennas 18 and m receive antennas 24. In this system 10, thechannel encoder 20 accepts input from the information source 12 andoutputs a coded stream of higher redundancy suitable for errorcorrection processing at the receiver 16. The encoded output stream ismodulated and distributed among the n antennas via a spatial modulator22. The signal received at each antenna 24 is a superposition of the ntransmitted signals corrupted by additive white Gaussian noise andmultiplicative fading. At the receiver 16, the signal r_(t) ^(j)received by antenna j at time t is given by $\begin{matrix}{r_{t}^{j} = {{\sqrt{E_{s}}{\sum\limits_{i = 1}^{n}{\alpha_{t}^{({ij})}c_{t}^{i}}}} + n_{t}^{j}}} & (1)\end{matrix}$

where {square root over (E_(s))} is the energy per transmitted symbol;α_(t) ^((ij)) is the complex path gain from transmit antenna i toreceive antenna j at time t; c_(t) ^(i) is the symbol transmitted fromantenna i at time t; n_(t) ^(j) is the additive white Gaussian noisesample for receive antenna j at time t. The symbols are selected from adiscrete constellation Ω containing 2^(b) points. The noise samples areindependent samples of zero-mean complex Gaussian random variable withvariance N₀/2 per dimension. The different path gains α_(t) ^((ij)) areassumed to be statistically independent.

The fading model of primary interest is that of a block flat Rayleighfading process in which the code word encompasses B fading blocks. Thecomplex fading gains are constant over one fading block but areindependent from block to block. The quasi-static fading model is aspecial case of the block fading model in which B=1.

The system 10 provides not one, but nm, potential communication linksbetween a transmitter 14 and a receiver 16, corresponding to eachdistinct transmit antenna 18/receive antenna 24 pairing. The space-timesystem 10 of the present invention is advantageous because it exploitsthese statistically independent, but mutually interfering, communicationlinks to improve communication performance.

2. Generalized Layering

In a layered space-time approach, the channel encoder 20 is composite,and the multiple, independent coded streams are distributed inspace-time in layers. The system 10 is advantageous because the layeringarchitecture and associated signal processing associated therewithallows the receiver 16 to efficiently separate the individual layersfrom one another and can decode each of the layers effectively. In suchschemes, there is no spatial interference among symbols transmittedwithin a layer (unlike the conventional space-time code designapproach). Conventional channel codes can be used while the effects ofspatial interference are addressed in the signal processor design. Whilethis strategy reduces receiver complexity compared to the non-layeredspace-time approach, significant gains are possible without unduecomplexity when the encoding, interleaving, and distribution oftransmitted symbols among different antennas are optimized to maximizespatial diversity, temporal diversity, and coding gain.

A layer is defined herein as a section of the transmission resourcesarray (i.e., a two-dimensional representation of all availabletransmission intervals on all antennas) having the property that eachsymbol interval within the section is allocated to at most one antenna.This property ensures that all spatial interference experienced by thelayer comes from outside the layer. A layer has the further structuralproperty that a set of spatial and/or temporal cyclic shifts of thelayer within the transmission resource array provides a partitioning ofthe transmission resource array. This allows for a simple repeated useof the layer pattern for transmission of multiple, independent codedstreams.

Formally, a layer in an n×l transmission resource array can beidentified by an indexing set L⊂I_(n)×I_(l) having the property that thet-th symbol interval on antenna a belongs to the layer if and only if(a, t)εL. Then, the formal notion of a layer requires that, if (a, t)εLand (a′, t′)εL, then either t≠t′ or a≠a′—i.e., that a is a function oft.

Now, consider a composite channel encoder γ consisting of n constituentencoders γ₁, γ₂, . . . , γ_(n) operating on independent informationstreams. Let γ_(i):y^(k) _(^(i)) →y^(N) _(^(i)) , so that k=k₁+k₂+ . . .+k_(n) and N=N₁+N₂+ . . . +N_(n). Then, there is a partitioning u=u₁|u₂|. . . |u_(n) of the composite information vector uεy^(k) into a set ofdisjoint component vectors u_(i), of length k_(i), and a correspondingpartitioning γ(u)=γ₁(u₁)|γ₂(u₂)| . . . |γ_(n)(u_(n)) of the compositecode word γ(u) into a set of constituent code words γ_(i)(u_(i)), oflength N_(i). In the layered architecture approach, the space-timetransmitter assigns each of the constituent code words γ_(i)(u_(i)) toone of a set of n disjoint layers.

There is a corresponding decomposition of the spatial formattingfunction that is induced by the layering. Let f_(i) denote the componentspatial formatting function, associated with layer L_(i), which agreeswith the composite spatial formatter f regarding the modulation andformatting of the layer elements but which sets all off-layer elementsto complex zero. Then

f(γ(u))=f ₁(γ₁(u ₁))+f ₂(γ₂(u ₂))+ . . . +f _(n)(γ_(n)(u _(n))).

3. Algebraic Space-Time Code Design

A space-time code may be defined as an underlying channel code Ctogether with a spatial modulator function f that parses the modulatedsymbols among the transmit antennas. It is well known that thefundamental performance parameters for space-time codes are (1)diversity advantage, which describes the exponential decrease of decodederror rate versus signal-to-noise ratio (asymptotic slope of theperformance curve in a log-log scale); and (2) coding advantage whichdoes not affect the asymptotic slope but results in a shift in theperformance curve. These parameters are related to the rank andeigenvalues of certain complex matrices associated with the basebanddifferences between two modulated code words.

Algebraic space-time code designs achieving full spatial diversity aremade possible by the following binary rank criterion for binary,BPSK-modulated space-time codes:

Theorem 1 (Binary Rank Criterion) Let be a linear n×l space-time codewith underlying binary code C of length N=nl where l≧n. Suppose thatevery non-zero code word ĉ is a matrix of full rank over the binaryfield . Then, for BPSK transmission over the quasi-static fadingchannel, the space-time code achieves full spatial diversity nm.

Proof: The proof is discussed in the above-referenced application Ser.No. 09/397,896.

Using the binary rank criterion, algebraic construction for space-timecodes is as follows.

Theorem 2 (Stacking Construction) Let M₁, M₂, . . . , M_(n) be binarymatrices of dimension k×l, l≧k, and let be the n×l space-time code ofdimension k consisting of the code word matrices${\hat{c} = \begin{bmatrix}{\underset{\_}{x}\quad M_{1}} \\{\underset{\_}{x}\quad M_{2}} \\\vdots \\{\underset{\_}{x}\quad M_{n}}\end{bmatrix}},$

where x denotes an arbitrary k-tuple of information bits and n≦l. Thensatisfies the binary rank criterion, and thus, for BPSK transmissionover the quasi-static fading channel, achieves full spatial diversitynm, if and only if M₁, M₂, . . . M_(n) have the property that

∀a ₁ , a ₂ , . . . , a _(n)ε:

M=a ₁ M ₁ ⊕a ₂ M ₂ ⊕ . . . ⊕a _(n) M _(n) is of full rank k unless a ₁=a ₂ = . . . =a _(n)=0.

Proof: The proof is discussed in the above-referenced application Ser.No. 09/397,896.

This construction is general for any number of antennas and, whengeneralized, applies to trellis as well as block codes. The BPSKstacking construction and its variations, including a similar versionfor QPSK transmission (in which case the symbol alphabet is ₄, theintegers modulo 4), encompass as special cases transmit delay diversity,hand-crafted trellis codes, rate 1/n convolutional codes, and certainblock and concatenated coding schemes. Especially interesting is theclass of rate 1/n convolutional codes with the optimal d_(free), most ofwhich can be formatted to achieve full spatial diversity.

In a layered architecture, an even simpler algebraic construction isapplicable to arbitrary signaling constellations. In particular, for thedesign of the component space-time code associated with layer L, we havethe following stacking construction using binary matrices for thequasi-static fading channel.

Theorem 3 (Generalized Layered Stacking Construction) Let L be a layerof spatial span n. Given binary matrices M₁, M₂, . . . , M_(n), ofdimension k×l, let C be the binary code of dimension k consisting of allcode words of the form g(x)=xM₁|xM₂| . . . |xM_(n), where x denotes anarbitrary k-tuple of information bits. Let f_(L) denote the spatialmodulator having the property that the modulated symbols μ(xM_(j))associated with xM_(j) are transmitted in the l/b symbol intervals of Lthat are assigned to antenna j. Then, as the space-time code in acommunication system with n transmit antennas and m receive antennas,the space-time code consisting of C and f_(L) achieves spatial diversitydm in a quasi-static fading channel if and only if d is the largestinteger such that M₁, M₂, . . . , M_(n) have the property that

∀a ₁ , a ₂ , . . . , a _(n) ε, a ₁ +a ₂ + . . . +a _(n) =n−d+1:

M=[a ₁ M ₁ a ₂ M ₂ . . . a _(n) M _(n)] is of rank k over the binaryfield.

Proof: The proof is discussed in the above-referenced application filedconcurrently herewith.

Corollary 4 Full spatial diversity nm is achieved if and only if M₁, M₂,. . . , M_(n) are of rank k over the binary field.

The natural space-time codes associated with binary, rate 1/n,convolutional codes with periodic bit interleaving are advantageous forthe layered space-time architecture as they can be easily formatted tosatisfy the generalized layered stacking construction. Theseconvolutional codes have been used for a similar application, that is,the block erasure channel. The main advantage of such codes is theavailability of computationally efficient, soft-input/soft-outputdecoding algorithms.

The prior literature on space-time trellis codes treats only the case inwhich the underlying code has rate 1/n matched to the number of transmitantennas. In the development of generalized layered space-time codedesign of the present invention, consider the more general case in whichthe convolutional code has rate greater than 1/n is considered. Thetreatment includes the case of rate k/n convolutional codes constructedby puncturing an underlying rate 1/n convolutional code.

Let C be a binary convolutional code of rate k/n with the usual transferfunction encoder Y(D)=X(D)G(D). In the natural space-time formatting ofC, the output sequence corresponding to Y_(j)(D) is assigned to the j-thtransmit antenna. Let F_(l)(D)=[G_(1,l)(D) G_(2,l)(D) . . .G_(n,l)(D)]^(T). Then, the following theorem relates the spatialdiversity of the natural space-time code associated with C to the rankof certain matrices over the ring [[D]] of formal power series in D.

Theorem 5 Let denote the generalized layered space-time code consistingof the binary convolutional code C, whose k×n transfer function matrixis G(D)=[F₁(D) F₂(D) . . . F_(n)(D)], and the spatial modulator f_(L) inwhich the output Y_(j)(D)=X(D)·F_(j)(D) is assigned to antenna j alonglayer L. Let v be the smallest integer having the property that,whenever a₁+a₂+ . . . +a_(n)=v, the k×n matrix [a₁F₁ a₂F₂ . . .a_(n)F_(n)] has full rank k over [[x]]. Then the space-time codeachieves d-level spatial transmit diversity over the quasi-static fadingchannel where d=n−v+1 and v≧k.

Proof: The proof is discussed in the above-referenced application filedconcurrently herewith.

Rate 1/n′ convolutional codes with n′<n can also be put into thisframework. This is shown by the following example. Consider the optimald_(free)=5 convolutional code with generators G₀(D)=1+D² andG₁(D)=1+D+D². In the case of two transmit antennas, it is clear that thenatural layered space-time code achieves d=2 level transmit diversity.

In the case of four transmit antennas, note that the rate 1/2 code canbe written as a rate 2/4 convolutional code with generator matrix:${G(D)} = {\begin{bmatrix}{1 + D} & 0 & {1 + D} & 1 \\0 & {1 + D} & D & {1 + D}\end{bmatrix}.}$

By inspection, every pair of columns is linearly independent over [[D]].Hence, the natural periodic distribution of the code across fourtransmit antennas produces a generalized layered space-time codeachieving the maximum d=3 transmit spatial diversity.

For six transmit antennas, the code is expressed as a rate 3/6 code withgenerator matrix: ${G(D)} = {\begin{bmatrix}1 & 0 & 1 & 1 & 1 & 1 \\D & 1 & 0 & D & 1 & 1 \\0 & D & 1 & D & D & 1\end{bmatrix}.}$

Every set of three columns in the generator matrix has full rank over[[D]], so the natural space-time code achieves maximum d=4 transmitdiversity.

Thus far, the design of generalized layered space-time codes thatexploit the spatial diversity over quasi-static fading channels has beendiscussed. The results obtained for generalized layered space-time codedesign, however, are easily extended to the more general block fadingchannel. In fact, in the absence of interference from other layers, thequasi-static fading channel under consideration can be viewed as a blockfading channel with receive diversity, where each fading block isrepresented by a different antenna. For the layered architecture with ntransmit antennas and a quasi-static fading channel, there are nindependent and non-interfering fading links per code word that can beexploited for transmit diversity by proper code design. In the case ofthe block fading channel, there is a total of nB such links, where B isthe number of independent fading blocks per code word per antenna. Thus,the problem of block fading code design for the layered architecture isaddressed by simply replacing parameter n by nB.

For example, the following “multi-stacking construction” is a directgeneralization of Theorem 3 to the case of a block fading channel. Inparticular, special cases of the multi-stacking construction are givenby the natural space-time codes associated with rate k/n convolutionalcodes in which various arms from the convolutional encoder are assignedto different antennas and fading blocks (in the same way that Theorem 5is a specialization of Theorem 3).

Theorem 6 (Generalized Layered Multi-Stacking Construction) Let L be alayer of spatial span n. Given binary matrices M_(1,1), M_(2,1), . . . ,M_(n,1), . . . , M_(1,B), M_(2,B), . . . , M_(n,B) of dimension k×l, letC be the binary code of dimension k consisting of all code words of theform

g(x)=xM _(1,1) |xM _(2,1) | . . . |xM _(n,1) | . . . |xM _(1,B) |xM_(2,B) | . . . |xM _(n,B),

where x denotes an arbitrary k-tuple of information bits, and B is thenumber of independent fading blocks spanning one code word. Let f_(L)denote the spatial modulator having the property that μ(xM_(j,v)) istransmitted in the symbol intervals of L that are assigned to antenna jin the fading block v.

Then, as the space-time code in a communication system with n transmitantennas and m receive antennas, the space-time code consisting of C andf_(L) achieves spatial diversity dm in a B-block fading channel if andonly if d is the largest integer such that M_(1,1), M_(2,1), . . . ,M_(n,B) have the property that

∀a _(1,1) , a _(2,1) , . . . , a _(n,B) ε, a _(1,1) +a _(2,1) + . . . +a_(n,B) =nB−d+1:

M=[a _(1,1) M _(1,1) a _(2,1) M _(2,1) . . . a _(n,B) M _(n,B)] is ofrank k over the binary field.

4. CONCLUSIONS

An algebraic approach to the design of space-time codes for layeredspace-time architectures has been formulated; and new code constructionshave been presented for the quasi-static fading channel as well as themore general block fading channel. It is worth noting that, in theabsence of interference from other layers, the fading channelexperienced by a given coded layer is equivalent to a block fadingchannel with receive diversity. Thus, the algebraic framework providedby the present invention is also useful for block fading channelswithout transmit diversity.

Although the present invention has been described with reference to apreferred embodiment thereof, it will be understood that the inventionis not limited to the details thereof. Various modifications andsubstitutions have been suggested in the foregoing description, andothers will occur to those of ordinary skill in the art. All suchsubstitutions are intended to be embraced within the scope of theinvention as defined in the appended claims.

What is claimed is:
 1. A method of generating a space-time code forencoding information symbols comprising the steps of: defining a binaryrank criterion such that is a linear n×l space-time code with underlyingbinary code C of length N=nl where l≧n, and a non-zero code word ĉ is amatrix of full rank over a binary field to allow full spatial diversitynm for n transmit antennas and m receive antennas; and generating binarymatrices M₁, M₂, . . . , M_(n) of dimension k×l, l≧k, being said n×lspace-time code of dimension k and comprising code word matrices${\hat{c} = \begin{bmatrix}{\underset{\_}{x}\quad M_{1}} \\{\underset{\_}{x}\quad M_{2}} \\\vdots \\{\underset{\_}{x}\quad M_{n}}\end{bmatrix}},$

wherein x denotes an arbitrary k-tuple of said information symbols andn≦l, said binary matrices M₁, M₂, . . . , M_(n) being characterized by∀a ₁ , a ₂ , . . . , a _(n)ε: M=a ₁ M ₁ ⊕a ₂ M ₂ ⊕ . . . ⊕a _(n) M _(n)is of full rank k unless a ₁ =a ₂ = . . . =a _(n)=0 to allow said codeto satisfy said binary rank criterion.
 2. A method as claimed in claim1, wherein said generating step is used for binary phase shift keyingtransmission over a quasi-static fading channel and achievessubstantially full spatial diversity nm.
 3. A method as claimed in claim1, wherein said code is selected from the group consisting of a trelliscode, a block code, a convolutional code, and a concatenated code.
 4. Amethod of generating a space-time code for encoding information symbolscomprising the steps of: defining L as a layer of spatial span n, and Cas a binary code of dimension k comprising code words having the formg(x)=xM₁|xM₂| . . . xM_(n), where M₁, M₂, . . . , M_(n) are binarymatrices of dimension k×l and x denotes an arbitrary k-tuple of saidinformation symbols; modulating said code words xM_(j) such that saidmodulated symbols μ(xM_(j)) are transmitted in the l/b symbol intervalsof L that are assigned to an antenna j (18); and generating a space-timecode comprising C and f_(L) having d be the largest integer such thatM₁, M₂, . . . , M_(n) have the property that ∀a ₁ , a ₂ , . . . , a _(n)ε, a ₁ +a ₂ + . . . +a _(n) =n−d+1: M=[a ₁ M ₁ a ₂ M ₂ . . . a _(n) M_(n)] is of rank k over the binary field to achieve spatial diversity dmin a quasi-static fading channel.
 5. A method as claimed in claim 4,wherein said binary matrices M₁, M₂, . . . , M_(n) are of rank k overthe binary field to achieve substantially full spatial diversity nm forn transmit antennas and m receive antennas.
 6. A method as claimed inclaim 4, further comprising the step of transmitting said space-timecodes at a rate b(n−d+1) bits per signaling interval for a generalizedlayered communication system having n transmit antennas, a signalingconstellation of size 2^(b), and component codes achieving d-levelspatial transmit diversity constellation.
 7. A method for generating aspace-time code for encoding information symbols comprising the stepsof: defining as a generalized layered space-time code comprising abinary convolutional code C having a k×n transfer function matrix ofG(D)=[F₁(D) F₂(D) . . . F_(n)(D)], and a spatial modulator f_(L) inwhich the output Y_(j)(D)=X(D)·F_(j)(D) is assigned to an antenna jalong a layer L; defining v as the smallest integer having the propertythat, whenever a₁+a₂+ . . . +a_(n)=v, the k×n matrix [a₁F₁ a₂F₂ . . .a_(n)F_(n)] has full rank k over [[x]]; and generating a space-time codeto achieve d-level spatial transmit diversity over the quasi-staticfading channel where d=n−v+1 and v≧k.
 8. A method generating aspace-time code for encoding information symbols comprising the stepsof: defining L as a layer of spatial span n and C as a binary code ofdimension k comprising code words of the form g(x)=xM _(1,1) |xM _(2,1)| . . . |xM _(n,1) | . . . |xM _(1,B) |xM _(2,B) | . . . |xM _(n,B)where M_(1,1), M_(2,1), . . . M_(n,1), . . . , M_(1,B), M_(2,B), . . . ,M_(n,B) are binary matrices of dimension k×l and x denotes an arbitraryk-tuple of information bits, and B as the number of independent fadingblocks spanning one code word; modulating said code words xM_(j,v) suchthat said modulated symbols μ(xM_(j,v)) are transmitted in the symbolintervals of L that are assigned to antenna j in a fading block v; andgenerating said space-time code in a communication system with ntransmit antennas and m receive antennas such that comprising C andf_(L) achieves spatial diversity dm in a B-block fading channel, d beingthe largest integer such that M_(1,1), M_(2,1), . . . , M_(n,B) have theproperty that ∀a _(1,1) , a _(2,1) , . . . , a _(n,B) ε, a _(1,1) +a_(2,1) + . . . +a _(n,B) =nB−d+1: M=[a _(1,1) M _(1,1) a _(2,1) M _(2,1). . . a _(n,B) M _(n,B)] is of rank k over the binary field.
 9. Anencoding apparatus in a layered space-time architecture comprising: aninput device for receiving information symbols; and a processing devicefor encoding said information symbols using a space-time code generatedby defining a binary rank criterion such that is a linear n×l space-timecode with underlying binary code C of length N=nl where l≧n and anon-zero code word ĉ is a matrix of full rank over a binary field toallow full spatial diversity nm for n transmit antennas and m receiveantennas, and by generating binary matrices M₁, M₂, . . . , M_(n) ofdimension k×l, l≧k, being said n×l space-time code of dimension k andcomprising code word matrices ${\hat{c} = \begin{bmatrix}{\underset{\_}{x}\quad M_{1}} \\{\underset{\_}{x}\quad M_{2}} \\\vdots \\{\underset{\_}{x}\quad M_{n}}\end{bmatrix}},$

wherein x denotes an arbitrary k-tuple of said information symbols andn≦l, said binary matrices M₁, M₂, . . . , M_(n) being characterized by∀a ₁ , a ₂ , . . . , a _(n)ε: M=a ₁ M ₁ ⊕a ₂ M ₂ ⊕ . . . ⊕a _(n) M _(n)is of full rank k unless a ₁ =a ₂ = . . . =a _(n)=0 to allow said codeto satisfy said binary rank criterion.
 10. An encoding apparatus asclaimed in claim 9, wherein said code is selected from the groupconsisting of a trellis code, a block code, a convolutional code, and aconcatenated code.
 11. An encoding apparatus in a layered space-timearchitecture comprising: an input device for receiving informationsymbols; and a processing device for encoding said information symbolsusing a space-time code generated by defining L as a layer of spatialspan n and C as a binary code of dimension k comprising code wordshaving the form g(x)=xM₁|xM₂| . . . |xM_(n), where M₁, M₂, . . . , M_(n)are binary matrices of dimension k×l and x denotes an arbitrary k-tupleof said information symbols, by modulating said code words xM_(j) suchthat said modulated symbols μ(xM_(j)) are transmitted in the l/b symbolintervals of L that are assigned to an antenna j, and by generating saidspace-time code comprising C and f_(L) wherein d is the largest integersuch that M₁, M₂, . . . , M_(n) have the property that ∀a ₁ , a ₂ , . .. , a _(n) ε, a ₁ +a ₂ + . . . +a _(n) =n−d+1: M=[a ₁ M ₁ a ₂ M ₂ . . .a _(n) M _(n)] is of rank k over the binary field to achieve spatialdiversity dm in a quasi-static fading channel.
 12. An encoding apparatusas claimed in claim 11, wherein said binary matrices M₁, M₂, . . . ,M_(n) are of rank k over the binary field to achieve substantially fullspatial diversity nm for n transmit antennas and m receive antennas. 13.An encoding apparatus in a layered space-time architecture comprising:an input device for receiving information symbols; and a processingdevice for encoding said information symbols using a space-time codegenerated by defining as a generalized layered space-time codecomprising a binary convolutional code C having a k×n transfer functionmatrix of G(D)=[F₁(D) F₂(D) . . . F_(n)(D)], and by spatial modulatingf_(L) in which the output Y_(j)(D)=X(D)·F_(j)(D) is assigned to anantenna j along a layer L, v being defined as the smallest integerhaving the property that, whenever a₁+a₂+ . . . +a_(n)=v, the k×n matrix[a₁F₁ a₂F₂ . . . a_(n)F_(n)] has full rank k over [[x]], and generatingsaid space-time code achieving d-level spatial transmit diversity overthe quasi-static fading channel where d=n−v+1 and v≧k.
 14. An encodingapparatus in a layered space-time architecture comprising: an inputdevice for receiving information symbols; and a processing device forencoding said information symbols using a space-time code generated bydefining L as a layer of spatial span n and C as a binary code ofdimension k comprising code words of the form g(x)=xM_(1,1) |xM _(2,1) |. . . |xM _(n,1) | . . . |xM _(1,B) |xM _(2,B) | . . . |xM _(n,B) whereM_(1,1), M_(2,1), . . . , M_(n,1), . . . , M_(1,B), M_(2,B), . . .M_(n,B) are binary matrices of dimension k×l and x denotes an arbitraryk-tuple of information bits, and B as the number of independent fadingblocks spanning one code word, modulating said code words xM_(j,v) suchthat said modulated symbols μ(xM_(j,v)) are transmitted in the symbolintervals of L that are assigned to antenna j in a fading block v, andgenerating said space-time code in a communication system with ntransmit antennas and m receive antennas such that comprising C andf_(L) achieves spatial diversity dm in a B-block fading channel, d beingthe largest integer such that M_(1,1), M_(2,1), . . . , M_(n,B) have theproperty that ∀a _(1,1) , a _(2,1) , . . . , a _(n,B) ε, a _(1,1) +a_(2,1) + . . . +a _(n,B) =nB−d+1: M=[a _(1,1) M _(1,1) a _(2,1) M _(2,1). . . a _(n,B) M _(n,B)] is of rank k over the binary field.